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Dmitriy Zanin

Publications and source records attributed to Dmitriy Zanin.

At least 55 records · Page 3Linked to original sources
arXiv2019

Rosenthal's inequalities: $Δ-$norms and quasi-Banach symmetric sequence spaces

Let $X$ be a symmetric quasi-Banach function space with Fatou property and let $E$ be an arbitrary symmetric quasi-Banach sequence space. Suppose that $(f_k)_{k\geq0}\subset X$ is a sequence of independent random variables. We present a necessary and sufficient condition on $X$ such that the quantity $$\Big\|\ \Big\|\sum_{k=0}^nf_ke_k\Big\|_{E}\ \Big\|_X$$ admits an equivalent characterization in terms of disjoint copies of $(f_k)_{k=0}^n$ for every $n\ge 0$; in particular, we obtain the deterministic description of $$\Big\|\ \Big\|\sum _{k=0}^nf_ke_k\Big\|_{\ell_q}\ \Big\|_{L_p}$$ for all $0<p,q<\infty,$ which is the ultimate form of Rosenthal's inequality. We also consider the case of a $Δ$-normed symmetric function space $X$, defined via an Orlicz function $Φ$ satisfying the $Δ_2$-condition. That is, we provide a formula for \lq\lq $E$-valued $Φ$-moments\rq\rq, namely the quantity $\mathbb{E}\big(Φ\big(\big\|(f_k)_{k\geq0} \big\|_E \big)\big)$, in terms of the sum of disjoint copies of $f_k, k\geq0.$

math.PR↗
arXiv2019

Logarithmic submajorization, uniform majorization and Hölder type inequalities for $τ$-measurable operators

We extend the notion of the determinant function $Λ$, originally introduced by T.Fack for $τ$-compact operators, to a natural algebra of $τ$-measurable operators affiliated with a semifinite von Neumann algebra which coincides with that defined by Haagerup and Schultz in the finite case and on which the determinant function is shown to be submultiplicative. Application is given to Hölder type inequalities via general Araki-Lieb-Thirring inequalities due to Kosaki and Han and to a Weyl-type theorem for uniform majorization.

math.FA↗
arXiv2019

Dilation invariant Banach limits

We study two subclasses of Banach limits: the one consisting of Banach limits which are invariant with respect of the Cesàro operator and another one consists of Banach limits which are invariant with respect to all dilations. We prove that the first is a proper subset of the second. We also show that these classes are at the maximal distance from the set ${\rm ext} \mathfrak B$ of all extreme points of the set of all Banach limits.

math.FA↗
arXiv2019

Logarithmic submajorisation and order-preserving linear isometries

Let $\mathcal{E}$ and $\mathcal{F}$ be symmetrically $Δ$-normed (in particular, quasi-normed) operator spaces affiliated with semifinite von Neumann algebras $\mathcal{M}_1$ and $\mathcal{M}_2$, respectively. We establish a noncommutative version of Abramovich's theorem \cite{A1983}, which provides the general form of normal order-preserving linear operators $T:\mathcal{E} \stackrel{into}{\longrightarrow} \mathcal{F}$ having the disjointness preserving property. As an application, we obtain a noncommutative Huijsmans-Wickstead theorem \cite{Huijsmans_W}. By establishing the disjointness preserving property for an order-preserving isometry $T:\mathcal{E} \stackrel{into}{\longrightarrow} \mathcal{F}$, we obtain the existence of a Jordan $*$-monomorphism from $\mathcal{M}_1$ into $\mathcal{M}_2$ and the general form of this isometry, which extends and complements a number of existing results. In particular, we fully resolve the case when $\mathcal{F}$ is the predual of $\mathcal{M}_2$ and other untreated cases in [Sukochev-Veksler, IEOT, 2018].

math.FA↗
arXiv2019

Local invariants of non-commutative tori

We consider a generic curved non-commutative torus extending the notion of conformally deformed non-commutative torus from \cite{Connes-Tretkoff}. In general, a curved non-commutative torus is no longer represented by a spectral triple, not even by a twisted spectral triple. Therefore, the geometry of this manifold is governed by a positive second order differential operator (Laplace-Betrami operator) rather than a first order differential operator (Dirac operator). For this manifold, we prove an asymptotic expansion of the heat semi-group generated by Laplace-Beltrami operator and provide an algorithm to compute the local invariants which appear as coefficients in the expansion. This allows to extend the results of \cite{Connes-Tretkoff}, \cite{Connes-Moscovici}, \cite{FaKh} (beyond conformal case and/or for multi-dimensional tori).

math.OA↗
arXiv2019

Simultaneous upper triangular forms for commuting operators in a finite von Neumann algebra

The joint Brown measure and joint Haagerup--Schultz projections for tuples of commuting operators in a von Neumann algebra equipped with a faithful tracial state are investigated, and several natural properties are proved for these. It is shown that the support of the joint Brown measure is contained in the Taylor joint spectrum of the tuple, and also in the ostensibly smaller left Harte spectrum. A simultaneous upper triangularization result for finite commuting tuples is proved and the joint Brown measure and joint Haagerup--Schultz projections are shown to be have well under the Arens multivariate holomorphic functional calculus of such a commuting tuple.

math.OA↗
arXiv2019

Noncommutative Geometry for Symmetric Non-Self-Adjoint Operators

We introduce the notion of a pre-spectral triple, which is a generalisation of a spectral triple $(\mathcal{A}, H, D)$ where $D$ is no longer required to be self-adjoint, but closed and symmetric. Despite having weaker assumptions, pre-spectral triples allow us to introduce noncompact noncommutative geometry with boundary. In particular, we derive the Hochschild character theorem in this setting. We give a detailed study of Dirac operators with Dirichlet boundary conditions on open subsets of $\mathbb{R}^d$, $d \geq 2$.

math.OA↗
arXiv2018

A $C^*$-algebraic approach to the principal symbol II

We introduce an abstract theory of the principal symbol mapping for pseudodifferential operators extending the results of a preceding paper and providing a simple algebraic approach to the theory of pseudodifferential operators in settings important in noncommutative geometry. We provide a variant of Connes' trace theorem which applies to certain noncommutative settings, with a minimum of technical preliminaries. Our approach allows us to consider in a operators with non-smooth symbols, and we demonstrate the power of our approach by extending Connes' trace theorem to operators with non-smooth symbols in three examples: the Lie group $\mathrm{SU}(2)$, noncommutative tori and Moyal planes.

math.OA↗
arXiv2018

The Connes character formula for locally compact spectral triples

A fundamental tool in noncommutative geometry is Connes' character formula. This formula is used in an essential way in the applications of noncommutative geometry to index theory and to the spectral characterisation of manifolds. A non-compact space is modelled in noncommutative geometry by a non-unital spectral triple. Our aim is to establish the Connes character formula for non-unital spectral triples. This is significantly more difficult than in the unital case and we achieve it with the use of recently developed double operator integration techniques. Previously, only partial extensions of Connes' character formula to the non-unital case were known. In the course of the proof, we establish two more results of importance in noncommutative geometry: an asymptotic for the heat semigroup of a non-unital spectral triple, and the analyticity of the associated $ζ$-function. We require certain assumptions on the underlying spectral triple, and we verify these assumptions in the case of spectral triples associated to arbitrary complete Riemannian manifolds and also in the case of Moyal planes.

math.OA↗
arXiv2017

Decomposibility and norm convergence properties in finite von Neumann algebras

We study Schur-type upper triangular forms for elements, T, of von Neumann algebras equipped with faithful, normal, tracial states. These were introduced in a paper of Dykema, Sukochev and Zanin; they are based on Haagerup-Schultz projections. We investigate when the s.o.t.-quasinilpotent part of this decomposition of T is actually quasinilpotent. We prove implications involving decomposability and strong decomposability of T. We show this is related to norm convergence properties of the sequence |T^n|^{1/n} which, by a result of Haagerup and Schultz, is known to converge in strong operator topology. We introduce a Borel decomposability, which is a property appropriate for elements of finite von Neumann algebras, and show that the circular operator is Borel decomposable. We also prove the existence of a thin-spectrum s.o.t.-quasinilpotent operator in the hyperfinite II_1-factor.

math.OA↗
arXiv2017

Trace theorem for quasi-Fuchsian groups

We complete the proof of the Trace Theorem in the quantized calculus for quasi-Fuchsian group which was stated and sketched, but not fully proved, on pp. 322-325 in the book "Noncommutative Geometry" of the first author.

math.OA↗
arXiv2017

Cwikel estimates revisited

In this paper, we propose a new approach to Cwikel estimates both for the Euclidean space and for the noncommutative Euclidean space.

math.OA↗
arXiv2017

Weak type operator Lipschitz and commutator estimates for commuting tuples

Let $f: \mathbb{R}^d \to\mathbb{R}$ be a Lipschitz function. If $B$ is a bounded self-adjoint operator and if $\{A_k\}_{k=1}^d$ are commuting bounded self-adjoint operators such that $[A_k,B]\in L_1(H),$ then $$\|[f(A_1,\cdots,A_d),B]\|_{1,\infty}\leq c(d)\|\nabla(f)\|_{\infty}\max_{1\leq k\leq d}\|[A_k,B]\|_1,$$ where $c(d)$ is a constant independent of $f$, $\mathcal{M}$ and $A,B$ and $\|\cdot\|_{1,\infty}$ denotes the weak $L_1$-norm. If $\{X_k\}_{k=1}^d$ (respectively, $\{Y_k\}_{k=1}^d$) are commuting bounded self-adjoint operators such that $X_k-Y_k\in L_1(H),$ then $$\|f(X_1,\cdots,X_d)-f(Y_1,\cdots,Y_d)\|_{1,\infty}\leq c(d)\|\nabla(f)\|_{\infty}\max_{1\leq k\leq d}\|X_k-Y_k\|_1.$$

math.OA↗
arXiv2016

Johnson-Schechtman inequalities for noncommutative martingales

In this paper we study Johnson-Schechtman inequalities for noncommutative martingales. More precisely, disjointification inequalities of noncommutative martingale difference sequences are proved in an arbitrary symmetric operator space $E(\mathcal{M})$ of a finite von Neumann algebra $\mathcal{M}$ without making any assumption on the Boyd indices of $E$. We show that we can obtain Johnson-Schechtman inequalities for arbitrary martingale difference sequences and that, in contrast with the classical case of independent random variables or the noncommutative case of freely independent random variables, the inequalities are one-sided except when $E=L_2(0,1)$. As an application, we partly resolve a problem stated by Randrianantoanina and Wu in \cite{NW}. We also show that we can obtain sharp $Φ$-moment analogues for Orlicz functions satisfying $p$-convexity and $q$-concavity for $1 \leq p \leq 2$, $q=2$ and $p=2$, $2< q < \infty$. This is new even for the classical case. We also extend and strengthen the noncommutative Burkholder-Gundy inequalities in symmetric spaces and in the $Φ$-moment case.

math.PR↗
arXiv2016

Banach limits and traces on $\mathcal L_{1,\infty}$

We introduce a new approach to traces on the principal ideal $\mathcal L_{1,\infty}$ generated by any positive compact operator whose singular value sequence is the harmonic sequence. Distinct from the well-known construction of J.~Dixmier, the new approach provides the explicit construction of every trace of every operator in $\mathcal L_{1,\infty}$ in terms of translation invariant functionals applied to a sequence of restricted sums of eigenvalues. The approach is based on a remarkable bijection between the set of all traces on $\mathcal L_{1,\infty}$ and the set of all translation invariant functionals on $l_\infty$. This bijection allows us to identify all known and commonly used subsets of traces (Dixmier traces, Connes-Dixmier traces, etc.) in terms of invariance properties of linear functionals on $l_\infty$, and definitively classify the measurability of operators in $\mathcal L_{1,\infty}$ in terms of qualified convergence of sums of eigenvalues. This classification has led us to a resolution of several open problems (for the class $\mathcal L_{1,\infty}$) from~\cite{CS}. As an application we extend Connes' classical trace theorem to positive normalised traces.

math.OA↗
arXiv2016

Singular traces and perturbation formulae of higher order

Let $H, V$ be self-adjoint operators such that $V$ belongs to the weak trace class ideal. We prove higher order perturbation formula $$τ\big(f(H+V)-\sum_{j=0}^{n-1}\frac{1}{j!}\frac{d^j}{dt^j} f(H+tV)\big|_{t=0}\big)=\int_{\mathbb{R}} f^{(n)}(t)\,dm_n(t),$$ where $τ$ is a trace on the weak trace class ideal and $m_n$ is a finite measure that is not necessarily absolutely continuous. This result extends the first and second order perturbation formulas of Dykema and Shripka, who generalised the Krein and Koplienko trace formulas to the weak trace class ideal. We also establish the perturbation formulae when the perturbation $V$ belongs to the quasi-Banach ideal weak-$L_n$ for any $n \geq 1$.

math.FA↗
arXiv2016

Krein's trace theorem revisited

We supply the first proof of Krein's Trace Theorem which does not use complex analysis. Our proof holds for~$σ$-finite von Neumann algebras $\mathcal{M}$ of type II and unbounded perturbations from the predual of~$\mathcal{M}$.

math.OA↗